Midterm1 - Mathematics Department, UCLA T. Richthammer...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Mathematics Department, UCLA T. Richthammer spring 09, midterm 1 Apr 27, 2009 Midterm 1: Math 170B Probability, Sec. 1 1. Let Z = X + Y , where X, Y are independent RVs with uniform distribution on [0 , 1]. (a) Calculate E ( Z ), V ( Z ) and the PDF of Z . (b) State for which of the results in (a) you really need the independence of X, Y . Answer: (a) E ( X ) = R 1 0 xdx = 1 2 and E ( X 2 ) = R 1 0 x 2 dx = 1 3 , i.e. V ( X ) = 1 3 - 1 4 = 1 12 . So E ( Z ) = E ( X ) + E ( Y ) = 1 and V ( Z ) = V ( X ) + V ( Y ) = 1 6 . Using the formula from the lecture the PDF f of Z is f ( z ) = R f X ( x ) f Y ( z - x ) dx = R 1 { 0 x 1 , 0 z - x 1 } dx = R 1 { max { 0 ,z - 1 }≤ x min { 1 ,z }} dx , which is = R z 0 dx = z for 0 z 1 and = R 1 z - 1 dx = 2 - z for 1 z 2, so f Z ( z ) = z 1 { 0 z 1 } + (2 - z )1 { 1 <z 2 } . Without using the formula from the lecture: For 0 c 2 we have F Z ( c ) = P ( Z c ) = λ 2 ( A c ), where A c is the subset of a square below the line y = c - x . For c 1 we have λ 2 ( A c ) = 1 2 c 2 , and for c 1 we have λ 2 ( A c ) = 1 - 1 2 (2 - c ) 2 . So f Z ( c ) = F ± Z ( c ) = c for c 1 and = 2 - c for c > 1, and we get the same answer as above. (b) Independence is needed for the variance and the PDF, but not for the expectation.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

Midterm1 - Mathematics Department, UCLA T. Richthammer...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online